Newspace parameters
| Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 448.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(139.948491417\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{6} - 2x^{5} - 1783x^{4} + 11524x^{3} + 591715x^{2} - 2562258x - 7402941 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{18}\cdot 23 \) |
| Twist minimal: | no (minimal twist has level 224) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{6} - 2x^{5} - 1783x^{4} + 11524x^{3} + 591715x^{2} - 2562258x - 7402941 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -253\nu^{5} - 5687\nu^{4} + 365540\nu^{3} + 5817616\nu^{2} - 95282919\nu - 955557225 ) / 14807088 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 1493\nu^{5} - 14897\nu^{4} - 2658052\nu^{3} + 28377616\nu^{2} + 729263919\nu - 1971137151 ) / 14807088 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4279\nu^{5} + 200021\nu^{4} + 1044628\nu^{3} - 113927536\nu^{2} - 5402176011\nu - 11395523637 ) / 14807088 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 291\nu^{5} + 5099\nu^{4} - 371252\nu^{3} - 2661712\nu^{2} + 92737849\nu - 147769161 ) / 616962 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 23479\nu^{5} - 46795\nu^{4} - 44477804\nu^{3} + 255190832\nu^{2} + 15769897269\nu - 52113887685 ) / 14807088 \)
|
| \(\nu\) | \(=\) |
\( ( -3\beta_{5} + 19\beta_{4} - 7\beta_{3} + 7\beta_{2} + 169\beta _1 + 443 ) / 1472 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 77\beta_{5} + 3\beta_{4} + 57\beta_{3} - 793\beta_{2} + 3513\beta _1 + 436731 ) / 736 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -6219\beta_{5} + 20435\beta_{4} - 6231\beta_{3} + 45791\beta_{2} + 151801\beta _1 - 5896837 ) / 1472 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 63311\beta_{5} - 34503\beta_{4} + 46955\beta_{3} - 737875\beta_{2} + 1362779\beta _1 + 240415681 ) / 368 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 10006839 \beta_{5} + 25609543 \beta_{4} - 7966891 \beta_{3} + 93398827 \beta_{2} + \cdots - 15777983281 ) / 1472 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −87.0931 | 0 | −237.119 | 0 | −343.000 | 0 | 5398.21 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −53.2431 | 0 | 192.808 | 0 | −343.000 | 0 | 647.829 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −4.11315 | 0 | 284.644 | 0 | −343.000 | 0 | −2170.08 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 20.5488 | 0 | −475.129 | 0 | −343.000 | 0 | −1764.75 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 21.5194 | 0 | 119.530 | 0 | −343.000 | 0 | −1723.92 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 86.3811 | 0 | −64.7341 | 0 | −343.000 | 0 | 5274.70 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(7\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 448.8.a.bc | 6 | |
| 4.b | odd | 2 | 1 | 448.8.a.bf | 6 | ||
| 8.b | even | 2 | 1 | 224.8.a.h | yes | 6 | |
| 8.d | odd | 2 | 1 | 224.8.a.e | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 224.8.a.e | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 224.8.a.h | yes | 6 | 8.b | even | 2 | 1 | |
| 448.8.a.bc | 6 | 1.a | even | 1 | 1 | trivial | |
| 448.8.a.bf | 6 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 16T_{3}^{5} - 9264T_{3}^{4} - 100112T_{3}^{3} + 13286592T_{3}^{2} - 121431168T_{3} - 728547264 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(448))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 16 T^{5} + \cdots - 728547264 \)
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| $5$ |
\( T^{6} + \cdots - 47842615296000 \)
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| $7$ |
\( (T + 343)^{6} \)
|
| $11$ |
\( T^{6} + \cdots + 47\!\cdots\!44 \)
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| $13$ |
\( T^{6} + \cdots + 28\!\cdots\!76 \)
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| $17$ |
\( T^{6} + \cdots + 13\!\cdots\!44 \)
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| $19$ |
\( T^{6} + \cdots - 79\!\cdots\!56 \)
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| $23$ |
\( T^{6} + \cdots + 65\!\cdots\!16 \)
|
| $29$ |
\( T^{6} + \cdots - 11\!\cdots\!76 \)
|
| $31$ |
\( T^{6} + \cdots - 31\!\cdots\!56 \)
|
| $37$ |
\( T^{6} + \cdots + 74\!\cdots\!24 \)
|
| $41$ |
\( T^{6} + \cdots - 59\!\cdots\!56 \)
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| $43$ |
\( T^{6} + \cdots - 85\!\cdots\!76 \)
|
| $47$ |
\( T^{6} + \cdots + 14\!\cdots\!84 \)
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| $53$ |
\( T^{6} + \cdots + 14\!\cdots\!36 \)
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| $59$ |
\( T^{6} + \cdots - 11\!\cdots\!96 \)
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| $61$ |
\( T^{6} + \cdots + 44\!\cdots\!16 \)
|
| $67$ |
\( T^{6} + \cdots - 53\!\cdots\!76 \)
|
| $71$ |
\( T^{6} + \cdots - 56\!\cdots\!64 \)
|
| $73$ |
\( T^{6} + \cdots - 92\!\cdots\!04 \)
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| $79$ |
\( T^{6} + \cdots + 38\!\cdots\!16 \)
|
| $83$ |
\( T^{6} + \cdots + 25\!\cdots\!96 \)
|
| $89$ |
\( T^{6} + \cdots + 15\!\cdots\!16 \)
|
| $97$ |
\( T^{6} + \cdots - 43\!\cdots\!36 \)
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